# Contents

## Idea

The special case of super Yang-Mills theory over a spacetime of dimension 4 and with $N = 2$ supersymmetry.

## Properties

### Moduli space of vacua

A speciality of $N=2$, $D = 4$ SYM is that its moduli space of vacua has two “branches” called the Coulomb branch and the Higgs branch. This is the content of what is now called Seiberg-Witten theory (Seiberg-Witten 94). Review includes (Albertsson 03, section 2.3.4).

### Confinement

While confinement in plain Yang-Mills theory is still waiting for mathematical formalization and proof (see Jaffe-Witten), N=2 D=4 super Yang-Mills theory is has been obsrved in (Seiberg-Witten 94).

### Reduction to $D = 3$ super Yang-Mills

By dimensional reduction on $\mathbb{R}^3 \times S^1$ families of $N = 2, D = 4$ SYM theories interpolate to N=4 D=3 super Yang-Mills theory. (Seiberg-Witten 96).

### Construction by compactification of 5-branes

$N=2$ super Yang-Mills theory can be realized as the worldvolume theory of M5-branes compactified on a Riemann surface (Klemm-Lerche-Mayr-Vafa-Warner 96, Witten 97, Gaiotto 09), hence as a compactifiction of the 6d (2,0)-superconformal QFT on the M5. This in particular gives a geometric interpretation of Seiberg-Witten duality in 4d in terms of the 6d 5-brane geometry.

Specifically the AGT correspondence expresses this relation in terms of the partition function of the theory and a 2d CFT on the Riemann surface on which the 5-brane is compactified. See at AGT correspondence for more on this.

gauge theory induced via AdS-CFT correspondence

M-theory perspective via AdS7-CFT6F-theory perspective
11d supergravity/M-theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^4$compactificationon elliptic fibration followed by T-duality
7-dimensional supergravity
$\;\;\;\;\downarrow$ topological sector
7-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS7-CFT6 holographic duality
6d (2,0)-superconformal QFT on the M5-brane with conformal invarianceM5-brane worldvolume theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surfacedouble dimensional reduction on M-theory/F-theory elliptic fibration
N=2 D=4 super Yang-Mills theory with Montonen-Olive S-duality invariance; AGT correspondenceD3-brane worldvolume theory with type IIB S-duality
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=2 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$, Donaldson theory

$\,$

gauge theory induced via AdS5-CFT4
type II string theory
$\;\;\;\;\downarrow$ Kaluza-Klein compactification on $S^5$
$\;\;\;\; \downarrow$ topological sector
5-dimensional Chern-Simons theory
$\;\;\;\;\downarrow$ AdS5-CFT4 holographic duality
N=4 D=4 super Yang-Mills theory
$\;\;\;\;\; \downarrow$ topological twist
topologically twisted N=4 D=4 super Yang-Mills theory
$\;\;\;\; \downarrow$ KK-compactification on Riemann surface
A-model on $Bun_G$ and B-model on $Loc_G$, geometric Langlands correspondence

## References

### General

The terminology “Coulomb branch” and “Higgs branch” first appears in

The dimensional reduction to $D = 3$ was first considered in

The confinement-phenomenon was observed in

Reviews of that include

For references on wall crossing of BPS states see the references given there.

### Introductions and surveys

• Davide Gaiotto, Recent progress in $N=2$ $4d$ field theory (2009) (pdf)

• Gregory Moore, Four-dimensional $N=2$ Field Theory and Physical Mathematics (arXiv:1211.2331)

• Greg Moore, Surface Defects and the BPS Spectrum of $4d$ $N=2$ Theories (pdf)

### Construction from 5-branes

$N=2$ $D=4$ SYM including its Seiberg-Witten theory (Seiberg-Witten 94) may be understood as being the compactification of the 6d (2,0)-superconformal QFT on the worldvolume of M5-branes on a Riemann surface: the Riemann surface is identified with the Seiberg-Witten curve of complexified coupling constants. This observation goes back to

The further observation that therefore the sewing of Riemann surfaces on which one compactifies the M5-brane yields a gluing operation on N=2 SYM theories is due to

The topological twisting of the compactification which is used around (2.27) there was previously introduced in section 3.1.2 of

and is discussed also for instance in section 5.1 of

(This is possibly also the mechanism behind the AGT correspondence, though the details obehind that statement seem to be unclear.)

A brief review of these matters is in (Moore 12, section 7). A formalization of the topological twist in perturbation theory formalized by factorization algebras with values in BV complexes is in section 16 of

For more on this see at topologically twisted D=4 super Yang-Mills theory.

An amplification of the relevance of this to the understanding of S-duality/electric-magnetic duality is in

and the resulting relation to the geometric Langlands correspondence is discussed in

The corresponding dual theory under AdS-CFT duality is discussed in

Discussion of construction of just N=1 D=4 super Yang-Mills theory this way is in

• Ibrahima Bah, Christopher Beem, Nikolay Bobev, Brian Wecht, Four-Dimensional SCFTs from M5-Branes (arXiv:1203.0303)

Revised on January 15, 2016 03:06:05 by Urs Schreiber (78.35.50.158)